Revisiting Hua-Marcus-Bellman-Ando inequalities on contractive matrices
type: Item
instance of: scholarly article
title: Revisiting Hua-Marcus-Bellman-Ando inequalities on contractive matrices
zbMATH Open document ID: 1163.15021
author: Changqing Xu
author: Fuzhen Zhang
author: Zhaodi Xu
published in: Linear Algebra and its Applications
publication date: 2009-02-25T00:00:00Z
review text: A complex \(n\times n\) matrix~\(M\) is strictly contractive if \(I-M^*M\) is positive definite. \textit{L.-K.~Hua} [Acta Math. Sinica~5, 463--470 (1955; Zbl 0066.26601), Am. Math. Soc. Transl., II. Ser.~32, 265--272 (1963; Zbl 0128.01704)] proved that if \(A\) and \(B\) are strictly contractive, then \[ I-B^*B+(A-B)^*(I-AA^*)^{-1}(A-B)= (I-B^*A)(I-A^*A)^{-1}(I-A^*B). \] He also proved certain determinantal inequalities. One of them is that if \(A_1,\dots,A_m\) are strictly contractive and \(k\geq n-1\), then the matrix~\(C=(c_{ij})\), \(c_{ij}=\det(I-A_i^*A_j)^{-k}\) is nonnegative definite. His work inspired further research by \textit{M.~Marcus} [Am. Math. Mon. 65, 266--269 (1958; Zbl 0083.00802)], \textit{R.~Bellman} [Duke Math.~J.~26, 485--490 (1959; Zbl 0087.01502)] and \textit{T.~Ando} [Linear and Multilinear Algebra~8, 347--352 (1980; Zbl 0438.15019)]. The present authors revisit these results with comments, analogs and extensions.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 5518765
zbMATH Keywords: contractive matrices
zbMATH Keywords: determinants
zbMATH Keywords: matrix inequalities
zbMATH Keywords: Hua's determinant inequality
zbMATH Keywords: Schur complements
reviewed by: Jorma K. Merikoski
MaRDI profile type: Publication
full work available at URL: j.laa.2007.11.011
OpenAlex ID: W2086371523
cites work: Hua—Marcus inequalities
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cites work: Matrix theory. Basic results and techniques
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DOI: 10.1016/J.LAA.2007.11.011
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description: scientific article; zbMATH DE number 5518765